Some inverse problems for a viscoelastic medium with a physically non-linear inclusion

A plane finite viscoelastic domain with a physically non-linear inclusion of arbitrary form is considered. The problem of finding those loads which, acting on the outer boundary of the domain, are such that they produce a specified uniform stress—strain state in the inclusion, is solved. Examples, in particular, of the optimal deformation and fracture of the inclusion under creep conditions, are considered.     

Inductive Limits of Area-Preserving Diffeomorphism Groups

A compact oriented surface with a given area form is considered. For each coordinate domain, we take the group of diffeomorphisms supported in this domain and preserving the area form. Finally, consider codimension 1 normal subgroups in these groups. For each pair of coordinate domains one of which contains the closure of the other, there is an obvious inclusion of the corresponding groups. We describe the inductive limits (amalgams) of these two families of groups.     

The non-linear Saint-Venant problem of the torsion,stretching and bending of a naturally twisted rod

Problems on large stretching, torsional and bending deformations of a naturally twisted rod, loaded with end forces and moments, are considered from the point of view of the non-linear three-dimensional theory of elasticity. Particular solutions of the equations of elastostatics are found, which are two-parameter families of finite deformations and which possess the property that, for these deformations, the initial system of three-dimensional non-linear equations reduces to a system of equations with two independent variables. The use of these equations enables one to reduce certain Saint-Venant problems for a naturally twisted rod to two-dimensional non-linear boundary-value problems for a planar domain in the form of the cross-section of a rod. Different formulations of the two-dimensional boundary-value problem for the cross-section are proposed, which differ in the choice of the unknown functions. A non-linear problem of the torsion and stretching of a circular cylinder with helical anisotropy, which is reduced to ordinary differential equations, is considered as a special case.     

Creeping flow of a second grade fluid in a corner

A variant of Taylor’s (1962) [23] scraper problem, in which, the lower plate rotates is considered. The non-linear partial differential equations governing the flow of a second grade fluid are modeled and solved by using the domain perturbation technique considering the angular velocity of the rotating plate as a small parameter. Also the rheology of the second grade fluid is examined by depicting the profiles of the velocity, stream function, pressure and stress fields.     

Mathematical and Numerical Simulation of Equilibrium of an Elastic Body Reinforced by a Thin Elastic Inclusion

A boundary value problem describing the equilibrium of a two-dimensional linear elastic body with a thin rectilinear elastic inclusion and possible delamination is considered. The stress and strain state of the inclusion is described using the equations of the Euler–Bernoulli beam theory. Delamination means the existence of a crack between the inclusion and the elastic matrix. Nonlinear boundary conditions preventing crack face interpenetration are imposed on the crack faces. As a result, problem with an unknown contact domain is obtained. The problem is solved numerically by applying an iterative algorithm based on the domain decomposition method and an Uzawa-type algorithm for solving variational inequalities. Numerical results illustrating the efficiency of the proposed algorithm are presented.     

Stability of a smooth flow in a Laval nozzle

A boundary-value problem for a non-linear second-order equation of mixed type in a cylindrical domain is considered. This problem simulates the development of small disturbances in a transonic flow of a chemical mixture in a Laval nozzle. The existence of a regular solution is proved with the help of a priori estimates for a corresponding linear problem and the contractive mapping theorem. The solution of the linear problem is constructed by the Galerkin method.     

Rotating disk flow and heat transfer through a porous medium of a non-Newtonian fluid with suction and injection

The steady flow of an incompressible viscous non-Newtonian fluid above an infinite rotating porous disk in a porous medium is studied with heat transfer. A uniform injection or suction is applied through the surface of the disk. Numerical solutions of the non-linear differential equations which govern the hydrodynamics and energy transfer are obtained. The effect of the porosity of the medium, the characteristics of the non-Newtonian fluid and the suction or injection velocity on the velocity and temperature distributions is considered. The inclusion of the three effects, the porosity, the non-Newtonian characteristics, and the suction or injection velocity together has shown some interesting effects.     

Fractional non-linear modelling of ultracapacitors

In this paper, it is demonstrated that an ultracapacitor exhibits a non-linear behaviour in relation to the operating voltage. A set of fractional order linear systems resulting from a frequency analysis of the ultracapacitor at various operating points is first obtained. Then, a non-linear model is deduced from the linear systems set, so that its Taylor linearization around the considered operating points (for the frequency analysis), produces the linear system set. The resulting non-linear model is validated on a Hybrid Electric Vehicle (HEV) application.     

带有可变库存费的变质性物品的部分延期供给的库存控制模型

传统的库存控制模型都视需求率为常数,在这篇文章中,放松了这个假定,研究了库存费的两种可能的变化:(i)库存费的变化率为存储时间的函数;(ii)库存费的变化率为库存量的函数.在模型中允许短缺发生且假定短缺部分延期供给,且在需求率线性依赖于库存水平的情形下,发展了两个变库存费的库存控制模型.     

Linear stabilization of the programmed motions of non-linear controlled dynamical systems under parametric perturbations

A non-linear controlled dynamical system that describes the dynamics of a broad class of non-linear mechanical and electromechanical systems (in particular, electromechanical robot manipulators) is considered. It is proposed that the real parameter vector of a non-linear controlled dynamical system belongs to an assigned (admissible) constrained closed set and is assumed to be unknown. The programmed motion of the non-linear controlled dynamical system and the programmed control that produces it are assigned (constructed) by using an estimate, that is, the nominal value of the parameter vector of the non-linear controlled dynamical system, which differs from its actual value. A procedure for synthesizing stabilizing control laws with linear feedback with respect to the state that ensure stabilization of the programmed motions of the non-linear controlled dynamical system under parametric perturbations is proposed. A non-singular linear transformation of the coordinates of the state space that transforms the original non-linear controlled dynamical system in deviations (from the programmed motion and programmed control) into a certain non-linear controlled dynamical system of special form, which is convenient for analysing and synthesizing laws for controlling the motion of the system, is constructed. A certain non-linear controlled dynamical system of canonical form is derived in the original non-linear controlled dynamical system in deviations. The transformation of the coordinates of the state space constructed and the Lyapunov function methodology are used to synthesize stabilizing control laws with linear feedback with respect to the state, which ensure asymptotic stability as a whole of the equilibrium position of the non-linear controlled dynamical system of canonical form and dissipativity “in the large” of the non-linear controlled dynamical system of special form and of the original non-linear controlled dynamical system in deviations. In the control laws synthesized, the formulae for the elements of their matrices of the feedback loop gains do not depend on the real parameter vector of the non-linear controlled dynamical system, and they depend solely on the constants from certain estimates that hold for all of its possible values from an assigned set. Estimates of the region of dissipativity “in the large” of the non-linear controlled dynamical system of special form and the original non-linear controlled dynamical system in deviations closed by the stabilizing control laws synthesized are given, and estimates for their limit sets and regions of attraction are presented.     

Parallel multilevel solvers for the cardiac electro-mechanical coupling

We develop a parallel solver for the cardiac electro-mechanical coupling. The electric model consists of two non-linear parabolic partial differential equations (PDEs), the so-called Bidomain model, which describes the spread of the electric impulse in the heart muscle. The two PDEs are coupled with a non-linear elastic model, where the myocardium is considered as a nearly-incompressible transversely isotropic hyperelastic material. The discretization of the whole electro-mechanical model is performed by Q1 finite elements in space and a semi-implicit finite difference scheme in time. This approximation strategy yields at each time step the solution of a large scale ill-conditioned linear system deriving from the discretization of the Bidomain model and a non-linear system deriving from the discretization of the finite elasticity model. The parallel solver developed consists of solving the linear system with the Conjugate Gradient method, preconditioned by a Multilevel Schwarz preconditioner, and the non-linear system with a Newton–Krylov-Algebraic Multigrid solver. Three-dimensional parallel numerical tests on a Linux cluster show that the parallel solver proposed is scalable and robust with respect to the domain deformations induced by the cardiac contraction.     

Problems of stability with respect to part of the variables

The Lyapunov-Malkin theorem on stability and (simultaneously) exponential asymptotic stability with respect to part of the variables in the linear approximation in critical cases (in Lyapunov's sense) has served as a point of departure for various previous results. These results are strengthened by relaxing all additional assumptions (other than continuity) regarding the coefficients of the linear part of the non-linear system under consideration. The result is extended to the problem of polystability with respect to part of the variables. In addition, a method for narrowing down the admissible domain of variation of “uncontrollable” variables is worked out as applied to problems of asymptotic stability with respect to part of the variables. Examples are considered.     

A convergence and existence result for differential-algebraic inclusions

A general class of multistep methods for the approximate solution of a differential-algebraic inclusion is considered. The convergence of these methods as well as the existence of a (generalized) solution of the inclusion is proved.     

Theorem about existence and uniqueness of continuous solution of nonlocal problem for nonlinear hyperbolic equation

The aim of the paper is to give a theorem about the existence and uniqueness of the continuous solution of a non-linear differential hyperbolic problem with a nonlocal condition in a bounded domain. The Banach theorem about the fixed point is used to prove the existence and uniqueness of the problem considered. The results obtained in this paper can be applied in the theory of elasticity with better effect than the analogous known result with the classical initial condition.     

Free magnetohydrodynamic flow and convection from a vertical spinning cone with cross-diffusion effects

A study of cross-diffusion effects on convection from a vertically spinning cone under the influence of an external magnetic field is considered. The non-linear two-point boundary value problem governing the flow is solved numerically. Two different types of surface heating, namely linear surface temperature (LST) and linear surface heat flux (LSHF) are considered. A parametric study addressing the effects of various flow parameters on the fluid properties, the skin friction, heat and mass transfer coefficients is given.     

Inclusion Theorems for Non-linear Difference Equations with Applications

For non-linear difference equations an inclusion theorem is set up which guarantees the existence of solutions in a given asymptotic stripe. This theorem is the limit case of an inclusion theorem for a boundary value problem in a finite interval. Different applications to rational difference equations of order two are given.     

Relationship Between Geršgorin‐type Theorems for Certain Subclasses of H‐Matrices

It is well known that the eigenvalue inclusion domain can be obtained by Geršgorin theorem. It is also known that the Geršgorin theorem is equivalent to the statement that each strictly diagonally dominant (SDD) matrix is nonsingular. Similarly, statements about nonsingularity of some classes of matrices, which are generalizations of SDD class, produce new Geršgorin‐type theorems for eigenvalue inclusion domain. In this paper we will consider some subclasses of H‐matrices and corresponding (Geršgorin‐type) eigenvalue inclusion theorems. Finally, we will establish relationship between given inclusion domains.     

非线性Kelvin-Helmholtz不稳定性

本文用导数展开法对液体薄层与亚音速气流接壤时的界面稳定性作非线性分析.文中考虑了液体的表面张力与体积力,故非线性的Rayleigh-Taylor不稳定性可作为特例而导出;液体与气体均不计粘性.虽然Nayfeh[1]曾算过这一情况,但其三阶方程有遗漏(如213页的式(2.29)).同时解也不自洽(如其一阶解(2.31)并不满足他的初始条件(2.20)),此外,在截止波数附近,对行波他并未考虑.本文弥补了这些,并得出了新的结论.